Sketch a graph of each of the following relations. Using the vertical line test, which of thes
is NOT a function?
x=0

Ask by Gibson Grant. in the United States

Jan 22,2025

Question

Sketch a graph of each of the following relations. Using the vertical line test, which of thes is NOT a function? x=0 $$ \begin{array}{l}2 x+3 y=6 \ y=0 \ y=x\end{array} $$

UpStudy AI · Accepted Answer

Let's analyze each of the given relations, sketch their graphs, and determine which one is **not** a function using the **vertical line test**.

### 1. $$ x = 0 $$
- **Graph:** This is a vertical line that lies on the y-axis.
  
  ![Graph of x=0](https://i.imgur.com/Z0xkqZd.png)
  
- **Vertical Line Test:** A vertical line intersects the graph at infinitely many points (for every value of $$ y $$). 
- **Conclusion:** **Not a function** because for $$ x = 0 $$, there are infinitely many $$ y $$-values.

### 2. $$ 2x + 3y = 6 $$
- **Rearranged Form:** Solve for $$ y $$:
  $$
  3y = -2x + 6 \
  y = -\frac{2}{3}x + 2
  $$
  
- **Graph:** This is a straight line with a slope of $$ -\frac{2}{3} $$ and a y-intercept at $$ (0, 2) $$.
  
  ![Graph of 2x + 3y = 6](https://i.imgur.com/4ZxPZ9k.png)
  
- **Vertical Line Test:** Any vertical line intersects the graph at exactly one point.
- **Conclusion:** **Is a function**.

### 3. $$ y = 0 $$
- **Graph:** This is the x-axis itself.
  
  ![Graph of y=0](https://i.imgur.com/MgFnAly.png)
  
- **Vertical Line Test:** Any vertical line intersects the graph at exactly one point.
- **Conclusion:** **Is a function**.

### 4. $$ y = x $$
- **Graph:** This is a straight line passing through the origin with a slope of 1.
  
  ![Graph of y=x](https://i.imgur.com/Qk9vVlK.png)
  
- **Vertical Line Test:** Any vertical line intersects the graph at exactly one point.
- **Conclusion:** **Is a function**.

### **Summary**

- **Not a Function:** $$ x = 0 $$
- **Functions:** $$ 2x + 3y = 6 $$, $$ y = 0 $$, $$ y = x $$

### **Final Answer**

All relations except the vertical line x = 0 are functions. The relation x = 0 is not a function.
The vertical line $$ x = 0 $$ is **not a function** because it fails the vertical line test. The other relations $$ 2x + 3y = 6 $$, $$ y = 0 $$, and $$ y = x $$ are all functions.

Sketch a graph of each of the following relations. Using the vertical line test, which of thes
is NOT a function?
x=0

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